SCALAR AND VECTOR MULTISTATIC RADAR DATAMODELS

By

Tegan Webster

A Thesis Submitted to the Graduate

Faculty of Rensselaer Polytechnic Institute

in Partial Fulfillment of the

Requirements for the Degree of

DOCTOR OF PHILOSOPHY

Major Subject: MATHEMATICS

Approved by theExamining Committee:

Margaret Cheney, Thesis Adviser

David Isaacson, Member

William Siegmann, Member

Eric Mokole, Member

Rensselaer Polytechnic InstituteTroy, New York

December 2012(For Graduation December 2012)

CONTENTS

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

ACKNOWLEDGMENT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2. Scalar Radar Data Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 Electromagnetic Wave Equation . . . . . . . . . . . . . . . . . 6

2.2.2 Matched Filtering . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.3 Classical Ambiguity Function . . . . . . . . . . . . . . . . . . 9

2.2.4 Multistatic Ambiguity Function . . . . . . . . . . . . . . . . . 13

2.2.5 Fourier Transform Convention . . . . . . . . . . . . . . . . . . 14

2.3 Comparison of Deterministic and Statistical Multistatic AmbiguityFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.1 Deterministic Data Model . . . . . . . . . . . . . . . . . . . . 15

2.3.2 Statistical Data Model . . . . . . . . . . . . . . . . . . . . . . 18

2.3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3.4 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.4 Extension of the Deterministic Model . . . . . . . . . . . . . . . . . . 29

2.4.1 Model for Data . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.4.2 Image Formation . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.4.3 Analysis of the Image: Ambiguity Function . . . . . . . . . . 34

2.4.4 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3. Polarimetric Radar Data Model . . . . . . . . . . . . . . . . . . . . . . . . 42

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.2.1 Plane Wave Solution to the Wave Equation . . . . . . . . . . 44

ii

3.2.2 Polarization of Electromagnetic Plane Waves . . . . . . . . . . 45

3.2.3 Scattering of Polarized Electromagnetic Plane Waves . . . . . 48

3.3 Problem Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.4 Data Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.4.1 The Potential Formulation . . . . . . . . . . . . . . . . . . . . 55

3.4.2 Green’s Function Solution to the Wave Equation . . . . . . . . 59

3.4.3 Expression for the Electric Field . . . . . . . . . . . . . . . . . 60

3.4.4 Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.4.5 Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.4.6 Reception . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.5 Image Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

3.5.1 Contributions from each Transmitter Assumed Separable . . . 93

3.5.2 Contributions from each Transmitter Assumed Nonseparable . 95

3.6 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

3.6.1 Simulation Parameters . . . . . . . . . . . . . . . . . . . . . . 97

3.6.2 Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

3.6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

4. Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . 119

LITERATURE CITED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

APPENDICES

A. Derivation of the Global Ambiguity Function . . . . . . . . . . . . . . . . . 131

A.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

A.1.1 Rayleigh Distribution . . . . . . . . . . . . . . . . . . . . . . . 131

A.1.2 Swerling II Target Model . . . . . . . . . . . . . . . . . . . . . 132

A.2 Output of the Matched Filter . . . . . . . . . . . . . . . . . . . . . . 133

A.3 Derivation of the Optimal Global Statistic . . . . . . . . . . . . . . . 135

A.3.1 Probability Density Under H0 . . . . . . . . . . . . . . . . . . 135

A.3.2 Probability Density Under H1 . . . . . . . . . . . . . . . . . . 137

A.3.3 Likelihood Ratio . . . . . . . . . . . . . . . . . . . . . . . . . 138

A.3.4 Optimal Global Statistic . . . . . . . . . . . . . . . . . . . . . 140

A.4 Global Ambiguity Function . . . . . . . . . . . . . . . . . . . . . . . 140

iii

B. Properties Concerning Differentiation of Certain Spatial Integrals . . . . . 144

B.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

B.2 Problem Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

B.3 Relation for Scalar Functions . . . . . . . . . . . . . . . . . . . . . . 145

B.4 Higher Order Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 147

B.5 Derivation of (B.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

B.6 Derivation of (B.2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

C. Complex Inverse Fourier Transform of Certain Quantities . . . . . . . . . . 154

C.1 Derivation of (3.119) . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

C.2 Derivation of (3.128) . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

D. Scattering from a Perfectly Electrically Conducting Flat Rectangular Plate 161

D.1 Problem Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

D.2 Physical Optics Approximation . . . . . . . . . . . . . . . . . . . . . 163

D.3 Derivation of the Scattered Electric Field Es(r) . . . . . . . . . . . . 164

D.4 Derivation of the Scattering Matrices . . . . . . . . . . . . . . . . . . 166

iv

LIST OF FIGURES

2.1 Classical ambiguity function of an up-chirp. . . . . . . . . . . . . . . . . 11

2.2 Classical ambiguity function of a 20 chip pseudo-random phase code. . . 12

2.3 Arrangement of target, transmitter, and receivers for cases 2.3(a)-(c). . 26

2.4 Deterministic MAF with equally spaced receivers and stationary target,zero velocity cut. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.5 Statistical MAF with equally spaced receivers and stationary target,zero velocity cut. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.6 Deterministic MAF with receivers spaced in a rough line and stationarytarget, zero velocity cut. . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.7 Statistical MAF with receivers spaced in a rough line and stationarytarget, zero velocity cut. . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.8 Deterministic MAF with receivers spaced in a rough line and targetvelocity 1.5 km/s, zero velocity cut. . . . . . . . . . . . . . . . . . . . . 28

2.9 Statistical MAF with receivers spaced in a rough line and target velocity1.5 km/s, zero velocity cut. . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.10 Classical ambiguity functions of an up-chirp, (a), and two 20 chip ran-dom polyphase codes, (b) and (c), plotted against velocity and timedelay on a dB scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.11 Beam Pattern Power, F̃ 2m(θ), with N=10. . . . . . . . . . . . . . . . . . 37

2.12 Arrangement of target, transmitters, and receivers for cases 2.12(a)-(c). 37

2.13 MIMO ambiguity function (MAF) for bistatic transmitter and receivercase. Each antenna has 10 elements and the transmitted waveform isan up-chirp. Cut at correct velocity. . . . . . . . . . . . . . . . . . . . . 38

2.14 MAF for bistatic transmitter and receiver case. Each antenna has 25elements and the transmitted waveform is an up-chirp. Cut at correctvelocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.15 MAF for one transmitter and four receiver case. Each antenna has a10 element beam pattern and the transmitted waveform is an up-chirp,(a) zero velocity cut and (b) velocity cut at correct velocity. . . . . . . . 39

v

2.16 MAF for one transmitter and four receiver case. Each antenna has a 10element beam pattern and the transmitted waveform is the phasecodein Figure 3.11(b), (a) zero velocity cut and (b) velocity cut at correctvelocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.17 MAF for the case of two transmitters and two receivers. Each antennahas a 10 element beam pattern and each transmitter emits a distinctrandom polyphase code. Velocity cut at correct velocity. . . . . . . . . . 40

3.1 Spatial evolution of monochromatic plane wave components. . . . . . . 45

3.2 Polarization ellipse in the h-v plane for a wave traveling into the pagein the k̂ direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.3 Linear, circular, and elliptical polarization states. . . . . . . . . . . . . . 47

3.4 Coordinate systems corresponding to the forward scatter alignment(FSA) convention (a) and backscatter alignment (BSA) convention (b). 48

3.5 Block diagram of the polarimetric radar problem. . . . . . . . . . . . . 53

3.6 An arbitrary dipole of diameter 2a and length 2L. . . . . . . . . . . . . 53

3.7 Plane wave components radiated from a vertically polarized dipole an-tenna. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.8 One bistatic pair of the multistatic system consisting of two dipole an-tennas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.9 A modified BSA coordinate system where the unit vectors and angleshave been renamed to better suit a multistatic geometry. . . . . . . . . 55

3.10 Local coordinate system used to define the radiation vector. . . . . . . . 64

3.11 Classical ambiguity functions of an up-chirp, (a), and two 20 chip ran-dom polyphase codes, (b) and (c), plotted against velocity and timedelay on a dB scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

3.12 Classical ambiguity functions of an up-chirp with 10 GHz center fre-quency and 1 MHz bandwidth (a), the up-chirp after it has been radi-ated from a long thin dipole (b), and after radiation and reception onthe same dipole (c). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

3.13 Classical ambiguity functions of an up-chirp with 100 MHz center fre-quency and 1 MHz bandwidth (a), the up-chirp after it has been radi-ated from a long thin dipole (b), and after radiation and reception onthe same dipole (c). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

3.14 Arrangement of target, transmitters, and receivers for cases 3.14(a)-(e). 101

vi

3.15 Chirp waveform, for scene Figure 3.14(a), flat rectangular scatterer, (a)zero velocity cut and (b) velocity cut at correct velocity. . . . . . . . . . 103

3.16 Phase coded waveform, for scene Figure 3.14(a), flat rectangular scat-terer, (a) zero velocity cut and (b) velocity cut at correct velocity. . . . 104

3.17 Chirp waveform, for scene Figure 3.14(b), flat rectangular scatterer,total image. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

3.18 Chirp waveform, for scene Figure 3.14(b), flat rectangular scatterer, HHimage (a) and VV image (b). . . . . . . . . . . . . . . . . . . . . . . . . 105

3.19 Phase coded waveform, for scene Figure 3.14(b), flat rectangular scat-terer, total image. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

3.20 Phase coded waveform, for scene Figure 3.14(b), flat rectangular scat-terer, HH image (a) and VV image (b). . . . . . . . . . . . . . . . . . . 106

3.21 Chirp waveform, for scene Figure 3.14(c), flat rectangular scatterer,total image. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

3.22 Phase coded waveform, for scene Figure 3.14(c), flat rectangular scat-terer, total image. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

3.23 Chirp waveform, for scene Figure 3.14(d), flat rectangular scatterer,total image. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

3.24 Phase coded waveform, for scene Figure 3.14(d), flat rectangular scat-terer, total image. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

3.25 Chirp waveform, for scene Figure 3.14(e), flat rectangular scatterer,total image. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

3.26 Phase coded waveform, for scene Figure 3.14(e), flat rectangular scat-terer, total image. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

3.27 Phase coded waveform, for scene Figure 3.14(e), flat rectangular scat-terer, total image. Transmitter composed of both a horizontally polar-ized dipole and vertically polarized dipole. . . . . . . . . . . . . . . . . 111

3.28 Chirp waveform, for scene Figure 3.14(b), complex scatterer, total image.112

3.29 Chirp waveform, for scene Figure 3.14(b), complex scatterer, HH image(a), HV image (b), VH image (c), and VV image (d). . . . . . . . . . . 113

3.30 Phase coded waveform, for scene Figure 3.14(b), complex scatterer, to-tal image. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

vii

3.31 Phase coded waveform, for scene Figure 3.14(b), complex scatterer, HHimage (a), HV image (b), VH image (c), and VV image (d). . . . . . . . 114

3.32 Phase coded waveform, for scene Figure 3.14(e), flat rectangular scat-terer, total image formed assuming the transmitter contributions arenonseparable. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

3.33 Chirp waveform, for scene Figure 3.14(e), flat rectangular scatterer,total image formed assuming the transmitter contributions are nonsep-arable. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

3.34 Phase coded waveform, for scene Figure 3.14(e), flat rectangular scat-terer, total image formed assuming the transmitter contributions arenonseparable. Transmitter composed of both a horizontally polarizeddipole and vertically polarized dipole. . . . . . . . . . . . . . . . . . . . 117

B.1 Spheres vs(r1) and vs(r2) with surfaces σs(r1) and σs(r2), respectively. . 146

C.1 Contour of integration used to derive (3.119). . . . . . . . . . . . . . . . 155

C.2 Contour of integration used to derive (3.128). . . . . . . . . . . . . . . . 158

D.1 Forward scatter approximation (FSA) coordinate system. . . . . . . . . 161

D.2 A flat rectangular plate oriented along the x-z plane with normal point-ing out of the page in the ŷ direction. . . . . . . . . . . . . . . . . . . . 162

viii

ACKNOWLEDGMENT

I would like to thank my primary thesis advisor, Professor Margaret Cheney, for her

guidance, support, and advice. Her enthusiasm for mathematics and cooperative

research has influenced me greatly. My mentor at the Naval Research Laboratory,

Dr. Eric Mokole, significantly guided the scope and subject matter of this thesis;

I appreciate his generous offer of time and research resources. I would next like to

thank Professor David Isaacson for his time and support as a member of my thesis

committee and instructor of the many engaging courses that I took with him during

my time at RPI. I would also like to thank Professor William Siegmann, for his

guidance during my experiences as an undergraduate student and graduate student

at Rensselaer Polytechnic Institute.

I would like to thank all of the other teachers and professors that have chal-

lenged and encouraged me including Mrs. Jonas, Mr. Hopkins, Mrs. Champagne,

Mrs. Davis, Mrs. Raszewski, Professor Kovacic, and Professor Kapila. I am also

extremely grateful to Dawnmarie Robens of the RPI graduate program for her con-

tinual assistance and advice.

At the Naval Research Laboratory Radar Division I have had the opportunity

to work with some really exceptional people. I would like to thank my supervisor

Dr. Aaron Shackelford for his mentoring and Dr. Jimmy Alatishe for his informative

discussions about radar hardware. I am especially grateful to Dr. Thomas Higgins

for his advice, feedback, and time consuming proofreading of this thesis.

I thank all of my talented and outrageous friends from the math department:

Ashley Baer, Jessica Jones, Analee Miranda, Peter Muller, Heather Palmeri, Joseph

Rosenthal, Katie Voccola, and any others that I am sure to have forgotten. Home-

work sessions, stress relieving workouts, and pizza are some of things that I have

learned are best shared with awesome people.

I lastly thank my family for their limitless support and encouragement.

This work was supported by the Naval Research Laboratory and the National

Defense Science and Engineering Graduate Fellowship.

ix

ABSTRACT

The aim of this thesis is to further the theory for multistatic imaging of moving

targets through the development and simulation of scalar and vector radar data

models and accompanying imaging operations. In the first part of the thesis we

investigate scalar representations of multistatic radar data. We begin by comparing

two different approaches for developing a multistatic ambiguity function (MAF),

a tool used to assess performance of the waveforms and geometry of a multistatic

radar system jointly. One approach is deterministic in nature, originating from the

scalar wave equation, and the other is statistical, relying on a Neyman-Pearson

defined weighting of received data. Although the two methods are fundamentally

different in formulation, they are shown to yield similar results. We then build on

the data model for the existing deterministically derived MAF with the inclusion of

antenna beam patterns by relating the current density on the radiating and receiving

antennas to a far-field spatial weighting factor. From this model we develop an

imaging formula in position and velocity that can be interpreted in terms of filtered

backprojection or matched filtering and a corresponding ambiguity function or point-

spread function. We use the resulting data model and MAF to examine scenarios

with various geometries and transmit waveforms and we show that the performance

of a multistatic system depends critically on the system geometry and transmitted

waveforms.

In the second part of the thesis we develop a vector multistatic data model

incorporating polarization and antenna effects from transmitters and receivers mod-

eled as long thin dipoles. We derive the model beginning with the potential formu-

lation of Maxwell’s equations and describe radiation from a transmitting antenna,

scattering from a moving target, and reception at a receiving antenna in both the

time and frequency domains. This model is developed from beginning to end with

the transmit waveform and scattering behavior of the target left arbitrary and we

obtain physical intuition, greater understanding and control of assumptions, and

the ability to carefully model the desired multistatic scenario by formulating our

x

data model from first principles. Following formulation of the data model we de-

rive two imaging operations that combine the data collected at each receiver, first

assuming that the contributions from all transmitters in the scene are separable

and then assuming that the contributions from all transmitting antennas cannot be

separated and must be treated as a unit. We then utilize the presented data model

and imaging operations to simulate multiple antenna geometries and transmission

schemes. Scattering behavior of the target is modeled with both a bistatic scattering

matrix based on physical optics for a perfectly electrically conducting flat rectan-

gular plate and a general complex scattering matrix. Simulations exhibit the angle

and polarization dependent scattering behavior and cross-polarization of the inci-

dent electric field consistent with the scattering models. The images formed under

both the separable and nonseparable assumptions are comparable when waveforms

with low cross-correlation are used. We approach the multistatic radar problem by

combining an electromagnetic data model with signal processing to obtain an image,

but the data model can also be used to generate high-quality data for a variety of

applications.

xi

CHAPTER 1

Introduction

Multistatic radar has been an area of intense research in recent years due to the in-

herent flexibility and importance of radar as an all-weather electromagnetic sensor

and the additional theoretical advantages of a multistatic radar system. Tradi-

tional monostatic radars radiate electromagnetic energy from an antenna, the field

propagates through space and is reflected in many directions from objects in the

environment, some of the reflected energy is received by the antenna of the radar,

and through processing techniques information about the intercepted objects is ex-

tracted. Radar systems can be used to detect and track targets, detect moving

targets in clutter rich environments, form images of stationary or moving targets or

scenes, recognize meteorological conditions, and classify targets. Although for some

of these applications other sensors can be used, such as electro-optical sensors, radar

is functional when these sensors fail perhaps in the dark of night or behind cloud

formations.

Multistatic radar systems differ from traditional monostatic systems, or bistatic

systems characterized by a separated transmitter and receiver, in that they consist

of multiple transmitters and/or receivers. A multistatic system consisting of mul-

tiple transmitters and one or more receivers can transmit multiple waveforms from

collocated or distributed antennas, possibly illuminating a larger area than a mono-

static system. The power of the transmitting antenna is a limiting factor on the

extent of a radar’s visibility; the use of multiple transmitting antennas to illuminate

an area may help overcome the physical limitations of amplifier power and antenna

aperture sizes. It is also possible to augment fielded systems with additional low-

power passive components to form a multistatic system, leveraging cooperative or

noncooperative signals of opportunity. These passive radar systems are well suited

to a variety of situations because they can be inexpensive and unobtrusive.

The design of a multistatic system, however, requires consideration of many

factors including the number, geometry, polarization of the transmitters and re-

1

2

ceivers, and the waveforms that will be transmitted from each radiating antenna. It

is important to ensure that all system components are coherent in time (that there is

a common clock available to all transmitters and receivers) and frequency. If either

of these condition are not met there will be some loss of information. The fusion of

data received from multiple sensors, whether coherent or incoherent, is another issue

inherent to multistatic radar. Ongoing theoretical research is necessary to address

the complexity of multistatic radar.

The aim of this thesis is to further the theory for multistatic imaging of mov-

ing targets through the development and simulation of scalar and vector radar data

models and accompanying imaging operations. In the first part of the thesis we

investigate scalar representations of multistatic radar data. We begin by comparing

two different approaches for developing a multistatic ambiguity function (MAF), a

tool used to assess performance of the waveforms and geometry of a multistatic radar

system jointly. One approach is derived with a deterministic signal model from an

imaging perspective and the other is derived for specific statistical target assump-

tions from a detection perspective. The deterministic MAF is formulated from the

scalar wave equation and describes radiation of the transmitted waveforms, scatter-

ing from a distribution of moving point-like targets, and reception at the receiving

antennas. The statistical MAF is developed by defining an optimal multistatic

detector corresponding to a Swerling II type target with fluctuating complex reflec-

tivity. Although the two methods are fundamentally different in formulation, the

corresponding numerical simulations are shown to yield similar results.

We then build on the data model for the existing deterministically derived

MAF with the inclusion of antenna beam patterns by relating the current density

on the radiating and receiving antennas to a far-field spatial weighting factor. From

this model we develop an imaging formula in position and velocity that can be inter-

preted in terms of filtered backprojection or matched filtering and a corresponding

ambiguity function or point-spread function. We use the resulting data model and

MAF to examine scenarios with various geometries and transmit waveforms and we

show that the performance of a multistatic system depends critically on the system

geometry and transmitted waveforms.

3

In the second part of the thesis we develop a vector multistatic data model

incorporating polarization and antenna effects from transmitters and receivers mod-

eled as long thin dipoles. We derive the model beginning with the potential formu-

lation of Maxwell’s equations and describe radiation from a transmitting antenna,

scattering from a moving target, and reception at a receiving antenna in both the

time and frequency domains. Following formulation of the data model we derive two

imaging operations that combine the data collected at each receiver, first assuming

that the contributions from all transmitters in the scene are separable and then as-

suming that the contributions from all transmitting antennas cannot be separated

and must be treated as a unit. We then utilize the presented data model and imag-

ing operations to simulate multiple antenna geometries and transmission schemes.

Scattering behavior of the target is modeled with both a bistatic scattering ma-

trix based on physical optics for a perfectly electrically conducting flat rectangular

plate and a general complex scattering matrix. Simulations exhibit the angle and

polarization dependent scattering behavior and cross-polarization of the incident

electric field consistent with the scattering models. The images formed under both

the separable and nonseparable assumptions are comparable when waveforms with

low cross-correlation are used.

This work is novel in that the model is developed from beginning to end with

the transmit waveform and scattering behavior of the target left arbitrary. We

obtain physical intuition, greater understanding and control of assumptions, and

the ability to model the desired multistatic scenario carefully by formulating our

data model from first principles. This work is also relevant because we combine an

electromagnetic data model with signal processing to obtain an image. Although

electromagnetics and signal processing are rich areas of study for radar applications,

the two fields are infrequently combined.

The remainder of this thesis is organized as follows. In Chapter 2 we in-

vestigate scalar representations of multistatic radar data from the perspective of

the multistatic ambiguity function (MAF). In Chapter 3 we formulate a full vector

model for multistatic radar data including the polarization and scattering of electro-

magnetic waves and two corresponding imaging operations that combine the data

4

collected at each receiver. In Chapter 4 we conclude with a summary of this thesis

work, a description of areas left for future research, and a discussion of the spe-

cific contributions to the field of multistatic radar modeling and imaging of moving

targets.

CHAPTER 2

Scalar Radar Data Model

2.1 Introduction

In this chapter we investigate scalar representations of multistatic radar data

from the perspective of the multistatic ambiguity function (MAF). The classical am-

biguity function (CAF) is a tool that is used to assess the performance of monostatic

radar waveforms; the MAF is an analogous tool for multistatic radar systems. Mul-

tistatic radar systems are characterized by the number and geometry of transmitters

and receivers, the choice of antennas, the transmitted waveforms, and the method

of fusing data received by multiple sensors. The added flexibility of a multistatic

radar system results in the need for developing the more complex MAF as a metric

for assessing the choice of waveforms and the system geometry jointly. Although

the CAF is primarily important for waveform design, we will show that the MAF

can be derived as part of an imaging operation in position and velocity.

Modeling and design of multistatic radar systems has been an area of sub-

stantial research in recent years. There has been theory developed for multistatic

moving target detection [1–6], multistatic imaging of a stationary scene [7–10], mul-

tistatic imaging of moving targets [11–16], and coherence of components of a mul-

tistatic system [17–19]. Multiple formulations of the multistatic ambiguity func-

tion have been presented in the literature and will be briefly described in Section

2.2.4 [4, 5, 11–14,20–25].

In Chapter 2 we first provide some background on the electromagnetic wave

equation, matched filtering, the classical ambiguity function, and the multistatic

ambiguity function. The rest of the chapter is broken into two sections. In Section

2.3 we formulate two multistatic ambiguity functions that are found in the literature,

one derived with a deterministic signal model from an imaging perspective and one

derived for specific statistical target assumptions from a detection perspective. The

deterministic MAF is formulated from the scalar wave equation and models radiation

of the transmitted waveforms, scattering from a distribution of moving point-like

5

6

targets, and reception at the receiving antennas. The statistical MAF is developed

by defining an optimal multistatic detector corresponding to a Swerling II type target

with fluctuating complex reflectivity. After presenting both models we compare the

mathematical expressions and corresponding numerical simulations. We show that

although the derivations of the two MAFs are quite different, numerical results are

comparable and in the case of a single transmitter both mathematical expressions

can be written in terms of the classical ambiguity function. In Section 2.4 we

build on the data model for the existing deterministically derived MAF with the

inclusion of antenna beam patterns by relating the current density on the radiating

and receiving antennas to a far-field spatial weighting factor. The formulation yields

a data model that is appropriate for narrowband waveforms in the case when the

targets are moving slowly relative to the speed of light. From this model we develop

an imaging formula in position and velocity that can be interpreted in terms of

filtered backprojection or matched filtering and a corresponding ambiguity function

or point-spread function. We show through simulations how the resulting MAF can

be used to investigate the impact of geometry and transmit waveforms on multistatic

system performance.

2.2 Background

2.2.1 Electromagnetic Wave Equation

Maxwell’s equations in the time domain

∇ ·D = ρ∇ ·B = 0

∇× E = −∂B∂t

∇×H = J + ∂D∂t

7

combined with the free space constitutive relations

D = �0EB = µ0H

yield

∇ · E = 1�0ρ (2.1)

∇ ·B = 0 (2.2)

∇× E = −∂B∂t

(2.3)

∇×B = µ0J + µ0�0∂E∂t

(2.4)

where D(r, t) is the electric displacement field, B(r, t) is the magnetic inductionfield, E(r, t) is the electric field, H(r, t) is the magnetic intensity or magnetic field,ρ(r, t) is the charge density, J (r, t) is the current density, �0 is the permittivity offree space, and µ0 is the permeability of free space. In free space ρ(r, t) = 0 and

J (r, t) = 0 so that (2.1) and (2.4) become

∇ · E = 0 (2.5)

and

∇×B = µ0�0∂E∂t, (2.6)

respectively.

We can obtain the electromagnetic vector wave equation from Maxwell’s equa-

tions under the free space assumption. We begin by taking the curl of both sides of

(2.3) and substituting (2.6) to obtain

∇×∇× E = −µ0�0∂2E∂t2

. (2.7)

8

We next apply the vector identity

∇× (∇×A) = ∇(∇ ·A)−∇2A (2.8)

to (2.7)

∇(∇ · E)−∇2E = −µ0�0∂2E∂t2

(2.9)

and recall (2.5) to obtain the wave equation

∇2E = µ0�0∂2E∂t2

(2.10)

or (∇2 − 1

c20

∂2

∂t2

)E(r, t) = 0 (2.11)

where c0 = (µ0�0)−1/2. We have explicitly included the temporal and spatial depen-

dences of the electric field. Through a similar process we can obtain the vector wave

equation for the magnetic field H. Each component of the electric field or magneticfield satisfies the scalar wave equation. In Chapter 2 we consider solutions to the

scalar wave equation and in Chapter 3 we consider the vector wave equation derived

from the potential formulation of Maxwell’s equations.

2.2.2 Matched Filtering

The matched filter is the optimal linear filter for maximizing the signal-to-

noise ratio (SNR) for a signal received in white noise. The impulse response of the

matched filter is given by

h(t) = s∗(−t)

where s(t) is the transmitted waveform. The signal that is transmitted, scattered

from a target, and received is assumed to have the form

srec(t) = ρs(t− τ) + n(t)

where ρ is the scattering strength of the target including range losses, τ = 2R/c0

is the two-way time delay, R is the distance from the antenna to the target, and

9

n(t) is white noise. The output of the matched filter is obtained by convolving the

received signal with the impulse response

η(t) = (h ∗ srec)(t) =∫ ∞−∞

s∗(−t′)srec(t− t′)dt′ =∫ ∞−∞

s∗(t′)srec(t+ t′)dt′, (2.12)

which is the correlation of s(t) and srec(t). The matched filter output SNR, SNRmf,

is given by

SNRmf =2E

N0

where E is the energy of the received signal and N0 is the unilateral power spectral

density of white noise.

2.2.3 Classical Ambiguity Function

The classical ambiguity function (CAF), or Woodard ambiguity function, is a

tool that is used to assess the performance of monostatic radar waveforms [26–30].

The CAF describes the matched filter output for targets at different distances R

and velocities v and is expressed as

χ(τ, fd) =

∫ ∞−∞

s(t)s∗(t+ τ)ei2πfdtdt (2.13)

in terms of time delay τ = 2R/c0 and Doppler frequency fd = 2v/c0 where a

positive range R corresponds to a target further in range than a reference target

and a positive v denotes an incoming target. Letting

Ψ(τ, fd) = |χ(τ, fd)|2, (2.14)

if the signal is normalized to unit energy such that∫ ∞−∞|s(t)|2dt = 1

then the maximum value of (2.14) is attained at the origin

Ψ(τ, fd) ≤ Ψ(0, 0) = 1 (2.15)

10

and the volume under the surface of (2.14) is given by∫ ∞−∞

∫ ∞−∞

Ψ(τ, fd)dτdfd = 1. (2.16)

If the signal is not normalized to unit energy then (2.15) and (2.16) are equal to

(2E)2 where E is the energy of the signal. Sometimes (2.13) is referred to as the

autocorrelation function and (2.14) the ambiguity function [28,30]. The signs of the

time delay and Doppler frequency in (2.13) may be reversed.

If we assume that the received signal input for the matched filter has the form

srec(t) = s(t)ei2πfdt

with zero time delay and Doppler frequency fd, then the output of the matched

filter is given by

η(t) = (h ∗ srec)(t) =∫ ∞−∞

s(t′)s∗(t′ − t)ei2πfdt′dt′ = χ(−t, fd) (2.17)

so that the matched filter output for a target with Doppler frequency fd is a time-

reversed version of (2.13) [30].

We will now examine the classical ambiguity functions for two standard wave-

forms, a linear chirp and a pseudo-random phase-coded waveform. The linear chirp

has linear frequency modulation (LFM), constant amplitude, pulse width T , and

bandwidth B that is swept either up or down over the duration of the pulse. The

linear chirp is defined as

schirp(t) = A rect(t/T ) cos(2πf0t+ παt2) (2.18)

where

rect(x) =

1 |x| ≤ 1/20 |x| > 1/2 ,f0 is the carrier frequency, A is the amplitude, and α = ±B/T is the LFM slopethat is positive for an up-chirp and negative for a down-chirp. The classical ambi-

11

guity diagram of an up-chirp waveform with 10 GHz carrier frequency, 20 µs pulse

width, and 1 MHz bandwidth is plotted in power on a dB scale in Figure 2.1. We

have plotted (2.14) for velocities ranging from −3 km/s to 3 km/s and time delaysranging from −20 µs to 20 µs relative to a reference target. Depending on the defi-nition of the ambiguity function, Figure 2.1 plots either values from the ambiguity

function (2.14) or the magnitude squared of values from the ambiguity function

(2.13). The ridge extending from negative velocity and time delay to positive veloc-

Velocity (km/s)

Tim

e D

eley

(µs)

!2 0 2

!15

!10

!5

0

5

10

15

!50

!40

!30

!20

!10

0

Figure 2.1: Classical ambiguity function of an up-chirp.

ity and time delay, with slope 1/α = 1.33 (km/s)/µs, reflects the Doppler tolerance

of LFM waveforms. This property is beneficial for detection of fast moving targets

because it is not necessary to implement multiple matched filters to cover the range

of possible Doppler shifts. The time (range) sidelobes are low across all Doppler

frequencies (velocities) as shown by the low ambiguity outside of the ridge in Figure

2.1. Nonlinear frequency modulated (NLFM) waveforms increase the rate of change

of frequency modulation (FM) near the ends of the pulse and decrease the rate of

change near the center, resulting in a waveform that does not require frequency

domain weighting to reduce time sidelobes. Symmetric FM results in a thumbtack-

like ambiguity function while asymmetric FM results in an ambiguity function that

is more ridge-like. Nonlinear FM waveforms are less Doppler tolerant than LFM

waveforms and thus are better suited to applications where the approximate target

12

velocity is known, such as tracking [30].

The pulse of a phase-coded waveform is subdivided into N sub-pulses, or chips,

of duration δ = T/N and a phase modulation is applied to each sub-pulse. There are

numerous phase modulation schemes, some resulting in ambiguity functions that re-

semble the ridged LFM ambiguity function and others that have a more thumbtack-

like appearance. Pseudo-random phase codes, or pseudo-noise (PN) codes, are phase

codes that consist of a sequence of chips with pseudo-random phases that are deter-

ministically generated by any of a variety of mechanisms. These codes may appear

random to an outside observer without prior knowledge of the code and the cross-

correlation between two different codes is low compared to other types of waveforms.

Pseudo-random codes are used in communications: each subscriber is assigned a

unique code by the base station and is able to correctly extract the relevant signals

from the collection of signals intended for all subscribers through matched filtering.

The classical ambiguity diagram of a 20 chip pseudo-random phase code with 10

GHz carrier frequency, 20 µs pulse width, and 1 MHz bandwidth is plotted in power

on a dB scale in Figure 2.2. We have again plotted (2.14) for velocities ranging

from −3 km/s to 3 km/s and time delays ranging from −20 µs to 20 µs relative toa reference target. The thumbtack ambiguity diagram reflects the Doppler sensitiv-

Velocity (km/s)

Tim

e D

eley

(µs)

!2 0 2

!15

!10

!5

0

5

10

15

!50

!40

!30

!20

!10

0

Figure 2.2: Classical ambiguity function of a 20 chip pseudo-random phase code.

ity of pseudo-random phase coded waveforms, which is beneficial for slowly moving

13

target applications or when knowledge of target speed is desired. It is evident from

examining Figures 2.1 and 2.2 that the time (range) sidelobes are higher for the PN

code than for the LFM waveform, which may hinder the detection of small targets

in the sidelobes of large scatterers.

In addition to the waveforms briefly described above there are also continu-

ous waveforms (CW) of a single amplitude and frequency. Time-frequency coded

waveforms consist of a pulse train where each pulse is at a different frequency.

When modulated waveforms are matched filtered with the transmitted signal

as described in Section 2.2.2, the range resolution is improved to the same level that

could be achieved by a shorter pulse; this phenomenon is called pulse compression.

The SNR is also increased through pulse compression, reducing the transmit power

required to detect a target at a given range. Consequently, if a system is peak-power-

limited, it is possible to transmit a longer waveform and increase the transmitted

energy without losing range accuracy [28]. This discussion is not meant to cover

the immense field of waveform design but rather to orient the reader with some

terminology and concepts.

2.2.4 Multistatic Ambiguity Function

The multistatic ambiguity function (MAF), or multiple-input-multiple-output

(MIMO) ambiguity function, is a tool used to assess the waveforms and geometry

of a multistatic radar system jointly. The MAF is determined by the waveform

choice for each transmitter, the multistatic geometry, and the method of fusing data

received by multiple sensors; consequently there are numerous possible formulations.

The bistatic ambiguity function is closely related to the classical ambiguity

function but considers bistatic ranges and velocities. Although there is no issue of

combining information from multiple sensors as in the case of the MAF, the resulting

ambiguity function is geometry dependent. The effect of system geometry on the

shape of the bistatic ambiguity function has been considered in [31].

The added complexity of multiple transmitters and receivers in a multistatic

radar system has led to multiple formulations of the MAF. A MAF is presented

in [21] for a nonfluctuating point-like target of constant velocity and closely spaced

14

transmitting and receiving sensors so that the relative Doppler frequencies observed

by each sensor are assumed identical. Other work has focused on developing MAFs

from the perspective of determining an optimal multistatic detector corresponding

to a target with statistically varying complex amplitude. An incoherent single trans-

mitter and multiple receiver system and a moving target are considered in [4,5] and

the work is extended for multiple transmitters in [20]. Coherent and incoherent pro-

cessing of single transmitter and multiple receiver systems are considered in [24] and

multiple transmitter and receiver systems in [23,25]. MAFs have also been derived

with deterministic signal models and a greater focus on wave propagation [11–14,22].

Although the classical and bistatic ambiguity functions are typically used for

waveform design, the intrinsic dependance on geometry and fusion of information

elevate the MAF to a construct that is more closely related to imaging. The MAF

indicates that imaging can be thought of as a detection problem in position and

velocity.

In this chapter we consider in detail the deterministically derived MAF pre-

sented in [11–14, 22] and the statistically derived MAF presented in [4, 5, 20] with

multiple transmitters and receivers. We also formulate an extended deterministic

MAF [16].

2.2.5 Fourier Transform Convention

Throughout this thesis we will adopt the convention that the Fourier transform

is given by

F (ω) = F{f(t)} =∫ ∞−∞

f(t)eiωt dt (2.19)

and the inverse Fourier transform by

f(t) = F−1{F (ω)} = 12π

∫ ∞−∞

F (ω)e−iωt dω. (2.20)

2.3 Comparison of Deterministic and Statistical Multistatic

Ambiguity Functions

In Section 2.3 we consider multistatic radar systems consisting of M trans-

mitters at position ym with m = {1, 2, . . . ,M} and N receivers at position zn with

15

n = {1, 2, . . . , N}. We assume that all transmitters and receivers are stationaryand that the target located at x may be stationary or moving with velocity v. We

present a deterministic signal model and corresponding MAF in Section 2.3.1 and a

statistical signal model and corresponding MAF in Section 2.3.2. In Section 2.3.3 we

discuss the mathematical expressions and underlying assumptions of both MAFs.

We then present numerical simulations in Section 2.3.4 and conclude the section.

2.3.1 Deterministic Data Model

The multistatic ambiguity function presented in [11–14, 22] is obtained as a

byproduct of an imaging method. The imaging method involves first developing a

mathematical model for the scattered field, using this model in a matched filter that

is applied at each receiver, and coherently summing the resulting filtered outputs

with appropriate weights for each transmitter-receiver pair.

We derive the deterministic data model for a bistatic pair consisting of the

mth transmitter at position ym and the nth receiver at position zn. The derivations

in [11,12,22] develop a mathematical model for data from a single isotropic source.

For the transmitter source located at ym we denote the waveform by sm(t), the

transmission time by −Tm, and the wavefield at time t and position x by ψ(t,x).We assume that away from the targets, the source wavefield satisfies the scalar wave

equation [∇2 − c−20 ∂2t

]ψ(t,x) = δ(x− ym)sm(t+ Tm), (2.21)

where c0 is the speed of light in a vacuum. We denote by qv the phase-space

distribution of target reflectivity. In other words, qv(x − vt) is the reflectivity, attime t, of those scatterers moving with velocity v that, at time t = 0, were located

at position x . We write the total wavefield ψ as the sum ψ = ψin +ψsc where ψin is

the incident field and ψsc is the scattered field. Under the Born (single-scattering)

approximation, we can think of the reflected incident field as providing a source∫qv(x− vt)dv ψinm(t,x) for the scattered field ψscm(t, zn) received at zn such that

[∇2 − c−20 ∂2t ]ψscm(t, zn) =∫qv(x− vt)dv ψinm(t,x). (2.22)

16

We use the free-space Green’s function and several changes of variables to obtain

ψscm(t, zn) =

∫δ(t− t′ − |x+ vt′ − zn|/c0)

4π|x+ vt′ − zn|

×∫qv(x)dv

sm(t′ + Tm − |x+ vt′ − ym|/c0)

4π|x+ vt′ − ym|dt′dx. (2.23)

and we write Rx,n = |x − zn|, Rx,n = x − zn, and R̂x,n = Rx,n/Rx,n. We assumea slowly moving target (i.e. |vt| � |x − zn|), which allows us to carry out the t′

integration in (2.23):

ψscm(t, zn) =

∫sm(αx,v(t−Rx,n/c0)−Rx,m/c0 + Tm)qv(x)

(4π)2Rx,nRx,mµx,vdx dv (2.24)

where

µx,v =1 + R̂x,n · v/c0,

αx,v =1− R̂x,m · v/c01 + R̂x,n · v/c0

≈ 1− (R̂x,m + R̂x,n) · v/c0

=1 + βx,v,

βx,v =− (R̂x,m + R̂x,n) · v/c0. (2.25)

The scattered field in (2.24) can be viewed as a sum of attenuated, time-delayed,

Doppler-scaled copies of the transmitted waveform.

For the case of multiple transmitters, we assume that each of the N receivers

can identify which part of the signal is from which transmitter. This identification

of source transmitter could perhaps be done by separating the transmissions in

frequency or code; this issue is left for the future.

We construct an image Iu(p) as an approximation of qv(x), the true phase

space distribution of scatterers moving at velocity v and located at position x at

time t = 0. This image is formed by matched filtering the weighted scattered field

with a time-delayed, Doppler-scaled version of the transmitted waveform and then

17

by summing over all transmitters and receivers:

Iu(p) = (4π)2

M∑m=1

N∑n=1

Rp,nRp,mµp,uαp,uJm,n

×∫s∗m(αp,u(t−Rp,n/c0)−Rp,m/c0 + Tm)ψscm(t, zn)dt.

(2.26)

Here the star denotes complex conjugation, the weights Rp,m, Rp,n and µp,u are

introduced to cancel the denominator of (2.24) when p = x and u = v, and Jm,n

denotes a geometry-dependent weighting function. This weighting function is left

undetermined in [12]. In the simulations below, we take Jm,n to be equal for all

pairs of transmitters and receivers.

In order to characterize the imaging system, we relate the image to the true

phase-space reflectivity:

Iu(p) =

∫K(p,u;x,v)qv(x)dx dv (2.27)

where

K(p,u;x,v) =M∑m=1

N∑n=1

Jm,nαp,uRp,nRp,mµp,uRx,nRx,mµx,v

×∫s∗m(αp,u(t−Rp,n/c0)−Rp,m/c0 + Tm)

× sm(αx,v(t−Rx,n/c0)−Rx,m/c0 + Tm)dt (2.28)

is the weighted multistatic ambiguity function (MAF), referred to as the point-

spread function in [11,12,22]. For point-like targets (qv(x) = δ(x−x0)δ(v−v0)) theweighted MAF K(p,u;x0,v0) is the phase-space image of that target distribution.

We recall that the classical (narrowband) radar ambiguity function for the

waveform radiated by the mth transmitter can be written as

Am(ω̃, τ) = e−iωmτ∫sm(t)s

∗m(t− τ)eiω̃t dt (2.29)

18

where ωm = 2πfm and fm is the carrier frequency of the transmitted waveform

sm(t). Additionally, we write

ω̃ = ωm(βp,u − βx,v) = 2πfm(βp,u − βx,v)

where ωmβp,u is the angular Doppler shift for position p and velocity u. Then, in

the narrowband approximation, the weighted multistatic ambiguity function (2.28)

can be rewritten in the form

K(p,u;x,v) =M∑m=1

N∑n=1

Rm,nAm (ω̃, τ) (2.30)

where

Rm,n =Jm,nRp,nRp,mµp,uαp,uRx,nRx,mµx,vαx,v

, (2.31)

ω̃ =2πfm(βp,u − βx,v), (2.32)

τ =αp,u(Rp,n −Rx,n)

c0+Rp,m −Rx,m

c0+

(1− αp,u

αx,v

)(Rx,m

c0− Tm

). (2.33)

Clearly, the simpler case of a single transmitter with multiple receivers reduces

(2.30) to

K(p,u;x,v) =N∑n=1

RnA (ω̃, τ)

with τ and ω̃ defined above. In this case, there is only one transmitted waveform

and the weighting Rn only depends on the receiver.

2.3.2 Statistical Data Model

The multistatic ambiguity function, or global ambiguity function, as presented

in [4, 5, 20] determines weights for receiver contributions from the Neyman-Pearson

optimal global statistic corresponding to a Swerling II target. A simple model is

assumed for the received signal, weights are applied to the matched filtered output

from each receiver, and the weighted contributions are added noncoherently. An

overview of the derivation is presented in Section 2.3.2 and a more detailed derivation

19

of the global ambiguity function not readily available in the literature is given in

Appendix A.

As in the deterministic model, we let ym denote the position of the mth trans-

mitter with transmit waveform sm(t) and zn denote the position of the nth receiver.

At each transmitter, we assume that a coherent processing interval (CPI) consists

of a single pulse of duration Tdm and energy Em so that

sm(t) =√

2Em

20

phase φn denotes an independent random variable uniformly distributed on [0, 2π].

The real and imaginary components of the noise nn(t) are zero-mean Gaussian with

equal variance, unilateral power spectral density N0n, and Rayleigh distributed en-

velope. The superscript a is used to denote a value corresponding to an actual target

while the superscript h is used to denote a value corresponding to a hypothesized

target.

At each receiver we perform standard matched filtering of the received data

with the expected composite received waveform. The expected, normalized, and

weighted composite waveform at the nth receiver, pn(t;~τhn , ~ω

hn), is specified by

pn(t;~τhn , ~ω

hn) =

1

Bn

M∑m=1

bm,nfm(t− τhm,n)eiωhm,nt, (2.36)

where

~τhn = [τh1,n, . . . , τ

hM,n]

T (2.37)

~ωhn = [ωh1,n, . . . , ω

hM,n]

T .

The coefficients

bm,n =Rp,m=1Rp,m

√EmEm=1

(2.38)

are derived for each transmitter-receiver pair from the bistatic radar equation, under

the assumptions that b1,n = 1 and PmGm ∼ Em where Pm is the power of the mth

transmitter and Gm is the gain of the mth transmitter. By inspection, bm,n depends

only on the transmitter. The normalization constant Bn in (2.36) is chosen so that∫ ∞−∞

pn(t;~τhn , ~ω

hn)p∗n(t;~τ

hn , ~ω

hn)dt = 1 (2.39)

and ∫ ∞−∞

pn(t;~τan , ~ω

an)p∗n(t;~τ

an , ~ω

an)dt = 1. (2.40)

The received signal at the nth receiver can then be rewritten using (2.34), (2.36),

21

and (2.38) to obtain

rn(t) =anµnRp,n

M∑m=1

√Em

Rp,mfm(t− τamn)eiω

am,nt + nn(t)

=anBnµn

√Em=1

Rp,m=1Rp,npn(t;~τ

an , ~ω

an) + nn(t) (2.41)

where µn is the compensation constant [20]. The output of the matched filter at the

nth receiver is given by

dn =

∣∣∣∣∫ ∞−∞

rn(t)√N0n

p∗n(t;~τhn , ~ω

hn) dt

∣∣∣∣ . (2.42)By the assumptions that the envelope of the noise and the amplitude of the

complex reflectivity of the target are both Rayleigh distributed, the output of the

matched filter, dn, will be Rayleigh distributed whether or not a target is present. We

recall that a Rayleigh distributed random variable A with parameter A0 will satisfy

E{A2} = 2A20 and so p(A) = R(A,√

12E{A2}

)where R denotes the Rayleigh

probability density function

R(A,A0) =A

A20exp

{− A

2

2A20

}. (2.43)

Under H0, we have E{d2n} = 1, whereas under H1, we haveE{d2n} = ρn2 + 1 whereρn is the signal-to-noise ratio at the n

th receiver given by

ρn =4A20nB

2nµ

2nEm=1

N0nR2p,m=1R2p,n

. (2.44)

Thus, the distribution of dn is given by

H0 : p(dn|H0) = R(dn,

√1

2E{d2n}

)= R

(dn,

√1

2

)(2.45)

H1 : p(dn|H1) = R(dn,

√1

2E{d2n}

)= R

(dn,

√1

2

(ρn2

+ 1))

. (2.46)

We treat the data from each of the receivers asN independent observations and

22

so the joint probability density of d = [d1, . . . , dN ] is the product of the individual

probability densities and the likelihood ratio can be written as

L(d) =

∏Nn=1R

(dn,√

12

(ρn2

+ 1))

∏Nn=1R

(dn,√

12

)∝ eD (2.47)

where

D =N∑n=1

ρnρn + 2

d2n (2.48)

is the optimal global statistic in the Neyman-Pearson sense.

We write the global ambiguity function [32], or multistatic ambiguity function,

as

Θ(Th, Ta,Ωh,Ωa) =N∑n=1

cnΘn(~τhn , ~τ

an , ~ω

hn, ~ω

an) (2.49)

where

Θn(~τhn , ~τ

an , ~ω

hn, ~ω

an) =

∣∣∣∣∫ ∞−∞

pn(t;~τan , ~ω

an)p∗n(t;~τ

hn , ~ω

hn) dt

∣∣∣∣2 (2.50)is the ambiguity function for the nth receiver in terms of the composite waveforms

corresponding to a true and hypothetical target and

Th = {τhm,n}M×N , Ta = {τam,n}M×NΩh = {ωhm,n}M×N , Ωa = {ωam,n}M×N .

The weights cn are defined subject to∑N

n=1 cn = 1,

Θ(Ta, Ta,Ωa,Ωa) = 1, (2.51)

and

Θ(Th, Ta,Ωh,Ωa) =1

KE{Ds} (2.52)

where K is a normalization constant and Ds is the global statistic when only signal

is present in the received data.

23

We solve for K and obtain the weights, cn, of the individual ambiguity func-

tions,

cn =

ρ2n2(ρn+2)∑Nk=1

ρ2k2(ρk+2)

, n = 1, ..., N. (2.53)

It follows that the global ambiguity function, or statistically derived multistatic

ambiguity function, is given by

Θ(Th, Ta,Ωh,Ωa) =N∑n=1

cnΘn(~τhn , ~τ

an , ~ω

hn, ~ω

an)

where

Θn(~τhn , ~τ

an , ~ω

hn, ~ω

an) =

∣∣∣∣∫ ∞−∞

pn(t;~τan , ~ω

an)p∗n(t;~τ

hn , ~ω

hn) dt

∣∣∣∣2 ,cn =

ρ2n2(ρn+2)∑Nk=1

ρ2k2(ρk+2)

,

ρn =4A20nB

2nµ

2nEm=1

N0nR2x,m=1R2x,n

,

and pn(t;~τan , ~ω

an) and pn(t;~τ

hn , ~ω

hn) are the composite waveforms corresponding to a

true and hypothetical target, respectively.

The global ambiguity function can be expressed in terms of actual and hy-

pothetical ranges and velocities, as stated in [20], but for ease of comparison with

the deterministic model, we write the global ambiguity function in terms of p,u,x,

and v, where x and v are respectively the actual vector position and velocity of the

target, and p and u correspond to a hypothetical vector position and velocity of the

target. The statistically derived multistatic ambiguity function can be rewritten as

Θ(p,u,x,v) =N∑n=1

cnΘn(p,u,x,v) (2.54)

24

where

τam,n =Rx,m +Rx,n

c0, τhm,n =

Rp,m +Rp,nc0

ωam,n = ωmβx,v, ωhm,n = ωmβp,u

with

βx,v =− (R̂x,m + R̂x,n) · v/c0,

βp,u =− (R̂p,m + R̂p,n) · u/c0.

A more detailed derivation of the global ambiguity function is given in Appendix A.

For the much simplified case of a single transmitter, (2.54) can be written as

Θ(p,u,x,v) =N∑n=1

cnΘ̃n(p,u,x,v)

=N∑n=1

cn

∣∣∣∣∫ ∞−∞

f(t− τan)f ∗(t− τhn )ei(ωan−ωhn)t dt

∣∣∣∣2=

N∑n=1

cn

∣∣∣∣∫ ∞−∞

f(t)f ∗(t− τ)eiω̃t dt∣∣∣∣2 (2.55)

by several changes of variables and with

τ =Rx,m=1 +Rx,n − (Rp,m=1 +Rp,n)

c0

ω̃ =ωc(βp,u − βx,v) = 2πfc(βp,u − βx,v), (2.56)

where ωc = 2πfc and fc is the carrier frequency of the waveform sent from the single

transmitter.

2.3.3 Discussion

Although the deterministic and statistical approaches in Sections 2.3.1 and

2.3.2 appear to have different goals (imaging versus detection), in fact these goals

are closely related: both approaches provide information about target position and

25

velocity. In particular, plotting the detection test statistic as a function of target

position and velocity produces an image [33].

The two approaches are based on fundamentally different underlying assump-

tions. The deterministic approach neglects multiple scattering, and uses an isotropic

model for target scattering. The statistical approach assumes a Swerling II tar-

get and the resulting received signal, after processing, is assumed to be Rayleigh-

distributed with zero-mean, Gaussian, equal-variance, real and imaginary compo-

nents.

Both approaches, however, apply weights to the filtered data from each re-

ceiver and in the case of a single transmitter both MAFs reduce to summing ordinary

bistatic narrowband radar ambiguity functions, with arguments adjusted for target

locations and velocities. The weighting is done somewhat differently in the two

approaches. In the deterministic approach, the weighting is due to purely geomet-

rical factors and the data corresponding to each individual bistatic pair is weighted

uniquely [34]. In the statistical approach, a statistical criterion is used to deter-

mine the appropriate weights, and the statistics are assumed to already incorporate

information about the relevant geometry and target radar cross section.

The two approaches also differ in their assumptions about coherency of the sys-

tem. The statistical approach assumes a noncoherent system, and consequently the

summation of ambiguity functions is noncoherent. The deterministic approach can

accommodate either a coherent or noncoherent system; the summation of ambiguity

functions would then be coherent or noncoherent as appropriate. In our derivation

and subsequent simulations we assume the deterministic approach is coherent.

2.3.4 Simulations

Simulation Parameters

In the following simulations, we use a complex up-chirp waveform of unit

amplitude with 10 GHz carrier frequency, 100 MHz sampling frequency, 100 µs

pulse width, and 5 MHz bandwidth. The spatial region of interest is a circle of 10

km radius.

26

Scenarios

Target

Transmitter

Receivers

15 km

10 km

(a)

ReceiversTarget

Transmitter

10 km

(b)

Receivers

1.5 km/s

Target

Transmitter

10 km

(c)

Figure 2.3: Arrangement of target, transmitter, and receivers for cases 2.3(a)-(c).

Results

Figures 2.4 and 2.5 correspond to the scene in Figure 2.3(a). Four receivers

are spaced at a radius of 15 km and angles of 45◦, 135◦, 225◦, and 315◦ from the

center of the scene and a transmitter is placed 15 km from the center of the scene

at 90◦. We make an arbitrary choice to locate the target at the center of the scene

to impose equal propagation loses at each receiver, and the target has constant zero

velocity. The figures display the zero velocity cut and are shown with a normalized

dB colorscale. The peak, identified by a white circle, is in the correct location for

both models, but the ambiguity is lower for the deterministic model in Figure 2.4.

Figures 2.6 and 2.7 correspond to the scene in Figure 2.3(b). Four receivers

are spaced at radius 8, 6, 5, and 7 km and angle 30◦, 55◦, 110◦, and 140◦ respectively

from the center of the scene and a transmitter at radius 10 km and angle 90◦. The

target is located at the center of the scene and has zero velocity. Due to nonuniform

spacing of the receivers, the propagation loses along each path will differ. Again,

the peak is in the correct location for both models but the ambiguity is lower in

Figure 2.6.

Figures 2.8 and 2.9 correspond to the scene in Figure 2.3(c). This is the same

scene as figures 2.6 and 2.7 except that the target has velocity 1.5 km/s in direction

270◦. The figures display the zero velocity cut as before and the peak is spatially

27

Figure 2.4: Deterministic MAF with equally spaced receivers and stationary target,zero velocity cut.

Figure 2.5: Statistical MAF with equally spaced receivers and stationary target,zero velocity cut.

Figure 2.6: Deterministic MAF with receivers spaced in a rough line and stationarytarget, zero velocity cut.

28

Figure 2.7: Statistical MAF with receivers spaced in a rough line and stationarytarget, zero velocity cut.

shifted in the direction opposite of the target velocity by about 300 m in both figures,

as would be expected. As in the zero velocity cases, the ambiguity is lower in Figure

2.8.

Figure 2.8: Deterministic MAF with receivers spaced in a rough line and targetvelocity 1.5 km/s, zero velocity cut.

2.3.5 Conclusion

Despite differences in the underlying assumptions of the deterministic and

statistical models, the derived multistatic ambiguity functions and simulations are

very similar. The deterministic MAF in Section 2.3.1 is derived from an imaging

point of view, whereas the statistical MAF in Section 2.3.2 is derived from a detection

point of view. The summations of the received data at each antenna can be either

coherent or noncoherent for the deterministic MAF and are noncoherent for the

29

Figure 2.9: Statistical MAF with receivers spaced in a rough line and target velocity1.5 km/s, zero velocity cut.

statistical MAF. The papers [4, 5, 20] are mainly focused on weight determination.

The resulting MAFs differ yet both are modifications of the same classical ambiguity

function.

The simulations show that the deterministically and statistically derived mul-

tistatic ambiguity functions provide comparable results for the zero-velocity case as

well as for nonzero velocity cuts. The deterministic model has slightly more pro-

nounced spatial localization than the statistical model, as would be predicted when

comparing a coherent system to a noncoherent system.

The comparison of the resulting MAFs and corresponding simulations attests

to the close relationship between detection and imaging, as observed in other works

[33], and encourages thinking of imaging as a detection problem at each point in

space and velocity.

2.4 Extension of the Deterministic Model

In Section 2.4 we build on the data model for the existing deterministically

derived MAF from Section 2.3.1 with the inclusion of antenna beam patterns by

relating the current density on the radiating and receiving antennas to a far-field

spatial weighting factor. We begin in Section 2.4.1 by formulating a data model that

incorporates radiation of the transmitted waveforms, scattering from a distribution

of moving point-like targets, and reception at the receiving antennas. From this

model we develop an imaging formula in position and velocity and a corresponding

30

ambiguity function or point-spread function in Sections 2.4.2 and 2.4.3. We present

numerical simulations in Section 2.4.4 and then conclude.

2.4.1 Model for Data

Model for wave propagation

We model wave propagation and scattering by the scalar wave equation [35]

for the wavefield ψ(t,x) due to a current density jm(t+ Tm,x−ym) transmitted attime −Tm from location ym:

[∇2 − c−2(t,x)∂2t ]ψ(t,x) = µ0∂tjm(t+ Tm,x− ym) . (2.57)

where µ0 denotes the vacuum magnetic permeability.

A single scatterer moving at velocity v corresponds to an index-of-refraction

distribution n2(x− vt):

c−2(t,x) = c−20 [1 + n2(x− vt)] , (2.58)

where c0 is the speed of light in vacuum. We write qv(x − vt) = c−20 n2(x − vt).To model multiple moving scatterers, we let qv(x− vt)dx dv be the correspondingquantity for the scatterers in the volume dx dv centered at (x,v), the spatial dis-

tribution, at time t = 0, of scatterers moving with velocity v. Consequently, the

scatterers in the spatial volume dx (at x) give rise to

c−2(t,x) = c−20 +

∫qv(x− vt)dv . (2.59)

We note that the physical interpretation of qv involves a choice of a time

origin. A choice that is particularly appropriate, in view of our assumption about

linear target velocities, is a time during which the wave is interacting with targets

of interest. This implies that the activation of the antenna at ym takes place at

a negative time which we have denoted in (2.57) by −Tm. The wave equation

31

corresponding to (2.59) is then[∇2 − c−20 ∂2t −

∫qv(x− vt)dv ∂2t

]ψ(t,x) = µ0∂tjm(t+ Tm,x− ym) . (2.60)

Model for transmitted field

The “incident” field ψinm(t,x) is the field that is generated by the transmitter

at position ym and propagates into an empty universe:

[∇2 − c−20 ∂2t ]ψinm(t,x) = µ0∂tjm(t+ Tm,x− ym) . (2.61)

We assume that the antenna is constructed to be sufficiently broadband so that the

source term can be written as the product

µ0∂tjm(t,x) = sm(t)fm(x) (2.62)

where sm(t) is the waveform transmitted from the antenna located at ym and fm(x)

is a spatial factor. We recall that according to the convention adopted in (2.20),

sm(t) can be written in terms of its inverse Fourier transform as

sm(t) =1

2π

∫e−iωtSm(ω)dω . (2.63)

The frequency-domain version of (2.61) is then

[∇2 + k2]Ψinm(ω,x) = e−iωTmSm(ω)fm(x− ym), (2.64)

where k = ω/c0 and we can solve (2.64) to obtain

Ψinm(ω,x) =

∫eik|x−zn|

4π|x− zn|e−iωTmSm(ω)fm(zn − ym)dz. (2.65)

We assume that the antenna is distant from the target, so that |x − ym| �|zn−ym| and |x−ym| � k|zn−ym|2. Consequently in (2.65) we make the far-fieldexpansion

eik|x−zn|

4π|x− zn|≈ e

ik|x−ym|

4π|x− ym|eik

̂(x−ym)·(ym−zn) (2.66)

32

thus obtaining

Ψinm(ω,x) ≈eik|x−ym|

4π|x− ym|e−iωTmSm(ω)

∫eik

̂(x−ym)·(ym−zn)fm(zn − ym)dzn

=eik|x−ym|

4π|x− ym|e−iωTmSm(ω)Fm(k ̂(x− ym)) (2.67)

where Fm denotes the spatial Fourier transform of fm. Fm represents the far-field

beam pattern of the transmitting antenna as a function of frequency. We assume

that the antenna is sufficiently broadband that Fm is independent of frequency over

the effective support of the waveform Sm(ω). Hence, we replace Fm(k ̂(x− ym))with Fm(x̂− ym) to reflect the dependence of the beam pattern solely on the angledetermined by the transmitter and observation positions ym and x, respectively.

Consequently, the transmitted field in the time domain can be expressed as

ψinm(t,x) =Fm(x̂− ym)4π|x− ym|

sm(t− |x− ym|/c0 + Tm). (2.68)

Model for scattered field

We can likewise model the scattered field that is received at zn using the

scalar wave equation under the Born (single-scattering) approximation with a source∫qv(x− vt)dv ∂2t ψinm(t,x) provided by the reflected incident field:

[∇2 − c−20 ∂2t ]ψscm(t, zn) =∫qv(x− vt)dv∂2t ψinm(t,x). (2.69)

Solving for the scattered field we obtain

ψscm(t, zn) =

∫δ(t− t′ −Rx,n(t′)/c0)

4πRx,n(t′)

∫qv(x)dv

× Fm(R̂x,m(t′))s̈m(t

′ + Tm −Rx,m(t′)/c0)4πRx,m(t′)

dt′dx (2.70)

whereRx,m(t′) = x+vt′−ym, Rx,m(t′) = |Rx,m(t′)|, and R̂x,m(t′) = Rx,m(t′)/Rx,m(t′).

We assume that the targets are moving slowly, so that |v|t′ and k|v|2t′2 aremuch smaller than |x− ym| or |x− zn| where k = ωmax/c0 and ωmax is the effectivemaximum angular frequency of the signal sm(t). Thus, we can replace Rx,m(t

′) and

33

R̂x,m(t′) with Rx,m(0) and R̂x,m(0) such that and Rx,m(0) = x − ym, Rx,m(0) =

|Rx,m(0)|, and R̂x,m(0) = Rx,m(0)/Rx,m(0). The scattered field in the slow-movercase can then be written [11]

ψsc,Sm (t, zn) =

∫Fm(R̂x,m(0))

(4π)2Rx,n(0)Rx,m(0)µx,v(0)s̈m [φ(t,x,v)] qv(x)dx dv (2.71)

where

φ(t,x,v) = αx,v (t−Rx,n(0)/c0)−Rx,m(0)/c0 + Tm (2.72)

with Doppler scale factor

αx,v =1− R̂x,m(0) · v/c01 + R̂x,n(0) · v/c0

(2.73)

and

µx,v(0) = 1 + R̂x,n(0) · v/c0. (2.74)

We assume the radar system is using a narrowband waveform of the form

sm(t) = s̃m(t) e−iωmt (2.75)

where s̃m(t) is slowly varying, as a function of t, in comparison with exp(−iωmt)and ωm is the carrier frequency for the transmitter at position ym. The scattered

field in the narrowband case becomes

ψsc,SNm (t, zn) =

∫ −ω2mFm(R̂x,m(0))(4π)2Rx,n(0)Rx,m(0)µx,v(0)

s̃m [φ(t,x,v)] e−iωmφ(t,x,v)qv(x)dx dv.

(2.76)

Model for received data

By an argument similar to the one used to obtain the transmission beam pat-

tern Fm(R̂x,m(0)), we obtain a receiver antenna beam pattern Fn(R̂x,n(0)) [35]. We

observe that Fn is dependent on the angle determined by the receiver and observa-

tion positions zn and x, respectively. This factor arises from modeling the reception

process that occurs at the receiving antenna, zn, as an integration of the scattered

34

field ψsc,SNm (t, zn) over the antenna with an appropriate weighting function and ap-

plying the far-field expansion as before. Consequently, we express the received data

as

dm,n(t) =

∫ −ω2mFm(R̂x,m(0))Fn(R̂x,n(0))(4π)2Rx,n(0)Rx,m(0)µx,v(0)

s̃m [φ(t,x,v)] e−iωmφ(t,x,v)qv(x)dx dv.

(2.77)

2.4.2 Image Formation

The corresponding imaging operation involves applying a weighted matched

filter and summing over all transmitters ym and receivers zn. The phase-space image

is given by the expression

I∞(p,u) =∑m

∑n

(4π)2Rp,n(0)Rp,m(0)µp,u(0)αp,u−ω2m

Jm,n(p,u)

×∫s̃∗m [φ(t,p,u)] e

iωmφ(t,p,u)dm,n(t)dt. (2.78)

Here the star denotes complex conjugation, and Jm,n is a geometrical factor [11]

that depends on the configuration of transmitters and receivers.

2.4.3 Analysis of the Image: Ambiguity Function

We obtain the narrowband MIMO ambiguity function (point-spread function)

of the imaging system, KNB∞ (p,u;x,v), by substituting (2.77) into (2.78)

I∞(p,u) =

∫KNB∞ (p,u;x,v)qv(x)d

3x d3v (2.79)

with

KNB∞ (p,u;x,v) =∑m

∑n

Fm(R̂x,m(0))Fn(R̂x,n(0))Jm,n(p,u)

×∫s̃∗m [φ(t,p,u)] s̃m [φ(t,x,v)] e

iωm[φ(t,p,u)−φ(t,x,v)] dt.

(2.80)

Through some manipulation the MIMO ambiguity function (MAF) reduces

35

to [11]

KNB∞ (p,u;x,v) =∑m

∑n

eiΦm,nFm(R̂x,m(0))Fn(R̂x,n(0))J̃m,n(p,u)Am(ω̃, τ),

(2.81)

where J̃m,n is a geometrical factor closely related to Jm,n above,

Am(ω̃, τ) = e−iωmτ

∫s̃∗m(t− τ) s̃m(t) eiω̃tdt (2.82)

is a version of the classical monostatic narrowband radar ambiguity function, which

is defined here to include a phase, with parameters

ω̃ = ωm(βp,u − βx,v) (2.83)

τ = [(Rp,m(0) +Rp,n(0))− (Rx,m(0) +Rx,n(0))]/c0, (2.84)

and

exp [iΦm,n] = exp [ iωm(βp,u − βx,v)(Rx,m(0)/c0 − Tm)]

× exp [−ikmβu(Rx,n(0)−Rp,n(0))] (2.85)

with km = ωm/c0 and βx,v = −(R̂x,m(0) + R̂x,n(0)

)·v/c0. The narrowband result

in (2.81) clearly exhibits the importance of the bistatic bisector vectors R̂x,m(0) +

R̂x,n(0) and R̂p,m(0) + R̂p,n(0). We observe that the MAF (2.81) is a weighted sum

of classical narrowband ambiguity functions.

2.4.4 Simulations

In the following simulations, we examine multiple geometries with moving

targets.

Simulation Parameters

We use a complex up-chirp and two 20-chip random polyphase codes with

10 GHz carrier frequency, 20 MHz sampling frequency, 20 µs pulse width, and 1

36

MHz bandwidth. The classical ambiguity functions (CAFs) for these waveforms are

shown in Figures 2.10(a)-2.10(c). The ridge-like CAF with low range lobes of the

up-chirp and thumbtack CAF with higher range side lobes shown in Figure 2.10 are

evident in our simulations of the MAF for various geometries.

Velocity (km/s)

Tim

e D

eley

(µs)

!2 0 2

!15

!10

!5

0

5

10

15

!50

!40

!30

!20

!10

0

(a)

Velocity (km/s)

Tim

e D

eley

(µs)

!2 0 2

!15

!10

!5

0

5

10

15

!50

!40

!30

!20

!10

0

(b)

Velocity (km/s)

Tim

e D

eley

(µ s)

!2 0 2

!15

!10

!5

0

5

10

15

!50

!40

!30

!20

!10

0

(c)

Figure 2.10: Classical ambiguity functions of an up-chirp, (a), and two 20 chiprandom polyphase codes, (b) and (c), plotted against velocity and time delay on adB scale.

The beam pattern used in these simulations is based on a N -element uniform

linear array with half-wavelength spacing and half-power beam width of approxi-

mately 10◦ for N = 10 and 4◦ for N = 25. It is expressed as a function of angle

by

F̃m(θ) =1

N

sin(Nπ2

sin(θ))

sin(π2

sin(θ)) , −π

2≤ θ ≤ π

2(2.86)

where F̃m(θ) = Fm(R̂x,m(0)) = Fn(R̂x,n(0)) and θ is measured relative to antenna

boresight. The boresight direction is assumed to be in the direction of the scene

37

center.

!90 !70 !50 !30 !10 10 30 50 70 90!30

!25

!20

!15

!10

!5

0

Angle from Look Direction (degrees)

Bea

m P

atte

rn P

ower

(dB

)

Figure 2.11: Beam Pattern Power, F̃ 2m(θ), with N=10.

Scenarios

1.5 km/s

Target4 km

TransmitterReceiver

3 km

(a)

TransmitterReceivers

1.5 km/s

Target3 km

(b)

1.5 km/s

Target4 km

ReceiversTransmitters

3 km

(c)

Figure 2.12: Arrangement of target, transmitters, and receivers for cases 2.12(a)-(c).

Results

Figures 2.13-2.16 display the MAF for various geometries depicted in Figure

2.12(a)-2.12(c) and are shown with a normalized dB colorscale. The region displayed

is a 6 km by 6 km square. In some plots, peaks are identified by white circles. The

38

antenna beam patterns of the transmitters are evident in the plots of the MAF

power.

Figures 2.13 and 2.14 correspond to the bistatic scene in Figure 2.12(a). A

transmitter is spaced at radius 4 km and angle 50◦ a receiver at radius 4 km and

angle 130◦. The transmitted waveform is an up-chirp. We make an arbitrary choice

to locate the target at the center of the scene to impose equal propagation loses at

each receiver in multistatic cases described below, and the target and has velocity

1.5 km/s in direction 270◦. In the corresponding velocity cut shown, the peak is

in the correct location. The 10 element beam pattern in Figure 2.13 and the 25

element beam pattern in Figure 2.14 is clearly visible.

Figure 2.13: MIMO ambiguity function (MAF) for bistatic transmitter and receivercase. Each antenna has 10 elements and the transmitted waveform is an up-chirp.Cut at correct velocity.

Figure 2.14: MAF for bistatic transmitter and receiver case. Each antenna has 25elements and the transmitted waveform is an up-chirp. Cut at correct velocity.

39

Figures 2.15 and 2.16 correspond to the scene in Figure 2.12(b). Four receivers

are spaced in a line with a transmitter in the center at radius 4 km and angle 90◦. An

up-chirp waveform is used in Figure 2.15 and the phase code from Figure 3.11(b) is

used in Figure 2.16. The target is located at the center of the scene and has velocity

1.5 km/s in direction 270◦. Figure 2.15(a) depicts the zero velocity cut in which

the target is nearly in the correct location but is spatially shifted in the direction

opposite of the target velocity by about 300 m, as would be expected from the

up-chirp CAF in 3.11(a). Figure 2.15(b) depicts the correct velocity cut in which

the target is in the correct location. Figure 2.16(a) depicts the zero velocity cut in

which the target is not visible, indicative of the doppler intolerance of phase codes.

Figure 2.16(b) depicts the correct velocity cut in which the target is in the correct

location.

(a) (b)

Figure 2.15: MAF for one transmitter and four receiver case. Each antenna hasa 10 element beam pattern and the transmitted waveform is an up-chirp, (a) zerovelocity cut and (b) velocity cut at correct velocity.

Figure 2.17 corresponds to the scene in Figure 2.12(c). Two receivers are

spaced at radius 4 km and angle 45◦ and 135◦ from the center of the scene and two

transmitters at radius 4 km and angle 75◦ and 105◦. The rightmost transmitter

emits the random polyphase code in Figure 3.11(b) and the other the phase code in

Figure 3.11(c). The target is located at the center of the scene and has velocity 1.5

km/s in direction 270◦. In the corresponding velocity cut shown, the peak is in the

correct location. In other velocity cuts, not presented here, the target is not visible.

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(a) (b)

Figure 2.16: MAF for one transmitter and four receiver case. Each antenna has a10 element beam pattern and the transmitted waveform is the phasecode in Figure3.11(b), (a) zero velocity cut and (b) velocity cut at correct velocity.

Figure 2.17: MAF for the case of two transmitters and two receivers. Each an-tenna has a 10 element beam pattern and each transmitter emits a distinct randompolyphase code. Velocity cut at correct velocity.

2.4.5 Conclusion

We have outlined the development of a linearized imaging theory that combines

the spatial, temporal, and spectral aspects of scattered waves, and also incorporates

antenna beam patterns.

This imaging theory is based on the general (linearized) expression we derived

for waves scattered from moving objects, which we model in terms of a distribution

in phase space. The expression for the scattered waves takes the form of a superpo-

sition of weighted, time delayed, and frequency shifted versions of the incident field;

consequently we form an image by applying a weighted matched filter.

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The effect of incorporating a transmit beam pattern is consistent with expec-

tations; the returns from the main lobe are greater than that from the side lobes.

The derived MAF has the capability of more closely approximating reality than the

previous foundational work [22].

Our simulations show that the information that can be obtained depends crit-

ically on the transmitted waveforms. In particular, the Dopper-intolerant polyphase

coded waveforms produce a MAF with good localization in velocity, as expected.

The up-chirp MAF, on the other hand, has low sidelobes, but can cause the target

image to focus at the wrong location and wrong velocity.

CHAPTER 3

Polarimetric Radar Data Model

3.1 Introduction

In Chapter 2 we investigated scalar representations of multistatic radar data

originating from the scalar wave equation. In this chapter we formulate a full vec-

tor model for multistatic radar data including the polarization and scattering of

electromagnetic waves. As in the previous chapter, we must address the fusion of

data from multiple transmitters and receivers, a natural consequence of considering

a multistatic system. In the vector case we must also address the representation of

the electromagnetic vector fields and the transformation of these fields that occurs

when the waves are scattered off of a target. The possible advantages of both mul-

tistatic systems and polarimetric systems encourage the formulation of a full vector

model for multistatic radar data.

As discussed in Chapter 1, multistatic systems have a number of theoretical

advantages, including the ability to transmit multiple waveforms from collocated

or distributed antennas, thus enabling interrogation of larger areas of interest due

to the geometry of the system. It may also be possible to augment fielded systems

with additional low-power passive components, forming a bistatic or multistatic sys-

tem [30,36]. The performance of a multistatic system is heavily dependent upon the

number, geometry, and polarization of the transmitters and receivers and the wave-

forms that will be transmitted. An appropriate model can be used to characterize

how these parameters impact performance for a particular environment and targets

of interest. A multistatic system must also ensure that all constituent antennas are

coherent in time, that there is a common clock available to all transmitters and

receivers, and frequency or risk a loss of information [17–19].

There has been significant work done to develop models for multistatic radar

systems and to address issues related to the design of a multistatic system. There

has been theory developed for multistatic moving target detection [1–6], multistatic

imaging of a stationary scene [7–10], multistatic imaging of moving targets [11–16],

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and co